Searching for strange attractor in wastewater flow

Stochastic Environmental Research and Risk Assessment

Volumn 15, Issue 5, 2001, Pages 399-413

  Kim, H S ^a   Yoon, Y N ^b   Kim, J H ^b,c  

^a SunMoon University   (South Korea)

^b Korea University   (South Korea)

^c Kwangju University   (South Korea)

Author keywords

BDS statistic; Chaos; Close returns; Correlation dimension; Embedding; Water treatment

Indexed keywords

EID: 0035540297     PISSN: 14363240     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004770100078     Document Type: Article

References (48)

1. Albano AM, Muench J, Schwartz C, Mees AI, Rapp PE (1988) Singular-value decomposition and the Grassberger-Procaccia algorithm. Physical Review A 38: 3017-3026
2. Angelbeck DI, Minkara RY (1996) Closure on "Strange attractors and chaos in wastewater flow" by D. I. Angelbec and R. Y. Minkara. J. Environ. Eng., ASCE, 122(3): 242-243
3. Angelbeck DI, Minkara RY (1994) Strange attractors and chaos in wastewater flow. J. Environ. Eng., ASCE 120(1): 122-137
4. Barnett KD (1993) On the estimation of the correlation dimension and its application to radar reflector discrimination. NASA Report 4564, DOT/FAA/RD-93/41
5. Bormann NE, Kincanon E (1996) Discussion on "Strange attractors and chaos in wastewater flow" by D. I. Angelbeck and R. Y. Minkara. J. Environ. Eng., ASCE, 12(3): 240
6. Brock WA, Dechert WD, Scheinkman JA, LeBaron B (1996) A test for independence based on the correlation dimension. Econometric Reviews 15(3): 197-235
7. Brock WA, Hsieh DA, Lebaron B (1991) Nonlinear dynamics, chaos, and instability: Statistical Theory and Economic Evidence. MIT Press, Cambridge
8. Brock WA (1986) Distinguishing random and deterministic systems. J. Economic Theory 40: 168-195
9. Broomhead DS, King GP (1986) Extracting qualitative dynamics from experimental data. Physcia D 20: 217-236
10. Butler D, Friedler E, Gatt K (1995) Characterising the quantity and quality of domestic wastewater inflows. Water Sci. Tech. 31(7): 13-24
11. Casdagli M, Eubank S, Farmer JD, Gibson J (1991) State space reconstruction in the presence of noise. Physica D 51: 52-98
12. Ding M, Grebogi C, Ott E, Sauer T, Yorke JA (1993) Estimating correlation dimension from a chaotic time series: when does plateau onset occur? Physica D 69: 404-424
13. Eckmann JP, Ruelle D (1992) Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. Physica D 56: 185-187
14. Fernandez G, Garbrecht J (1996) Discussion on "Strange Attractors and Chaos in Wastewater Flow" by D. I. Angelbeck and R. Y. Minkara. J. Environ. Eng., ASCE, 12(3): 240-242
15. Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Physical Review A 55: 1134-1140
16. Gilmore CG (1993) A new test for chaos. J. Economic Behavior and Organization 22: 209-237
17. Graf KE, Elbert T (1990) Dimensional analysis of the waking EEG. In: Basar D (ed.) Chaos in Brain Function. Springer-Verlag, New York
18. Grassberger P (1990) An optimized box-assisted algorithm for fractal dimensions. Physical Letters A 148(1,2): 63-68
19. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D7: 153-180
20. Holzfuss J, Mayer-Kress G (1986) An approach to error-estimation in the application of dimension algorithms. In: Mayer-Kress G (ed.) Dimensions and Entropies in Chaotic Systems. Springer-Verlag, New York
21. Jeong GD, Rao AR (1996) Chaos characteristics of tree ring series. J. Hydrology 182: 239-257
22. Kim HS, Eykholt R, Salas JD (1998) Delay time window and plateau onset of the correlation dimension for small data sets. Physical Review E 58(5): 5676-5682
23. Kim HS, Eykholt R, Salas JD (1999) Nonlinear dynamics, delay times, and embedding windows. Physica D 127: 48-60
24. Kim HS, Salas JD, Eykholt R (2001) Nonlinear determinism in daily streamflows. J. Hydrologic Eng., ASCE, Submitted
25. Kugiumtzis DB (1996) State space reconstruction parameters in the analysis of chaotic time series - the role of the time window length. Physica D 95: 13-28
26. Lorenz EN (1963) Deterministic nonperiodic flow. J. Atmospheric Sci. 20: 130-141
27. Martinerie JM, Albano AM, Mees AI, Rapp PE (1992) Mutual information, strange attractors, and the optimal estimation of dimension. Physical Review A 45: 7058-7064
28. Mees AI, Rapp PE, Jennings LS (1987) Singular-value decomposition and embedding dimension. Physical Review A 36(1): 340-346
29. Mindlin GB, Gilmore R (1992) Topological analysis and synthesis of chaotic time series. Physica D 58: 229-242
30. Mindlin GB, Solari HG, Natuello MA, Gilmore R, Hou XJ (1991) Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction. J. Nonlinear Sci. 1: 147-173
31. Mindlin GB, Hou XJ, Solari HG, Gilmore R, Tufillaro NB (1990) Classification of strange attractors by integers. Physical Review Letters 64(20): 2350-2353
32. Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Physical Review Letters 45(9): 712-716
33. Porporato A, Ridolfi L (1997) Nonlinear analysis of river flow time sequences. Water Resources Research 33(6): 1353-1367
34. Puente CE, Obregon N (1996) A deterministic geometric representation of temporal rainfall: results for a storm in Boston. Water Resources Research 32(9): 2825-2839
35. Ramsey JB, Yuan HJ (1990) The statistical properties of dimension calculations using small data sets. Nonlinearity 3: 155-175
36. Ramsey JB, Yuan HJ (1989) Bias and error bars in dimension calculations and their evaluation in some simple models. Physics Letters A 134: 287-297
37. Rodriguez-Iturbe I, Power BFD, Sharifi MB, Georgakakos KP (1989) Chaos in rainfall. Water Resources Res. 25(7): 1667-1675
38. Rosenstein MT, Collins JJ, De Luca CJ (1994) Reconstruction expansion as a geometry-based framework for choosing proper delay times. Physica D 73: 82-98
39. Sangoyomi TB, Lall U, Abarbanel HDI (1996) Nonlinear dynamics of the Great Salt Lake: dimension estimation. Water Resources Res. 32(1): 149-159
40. Sauer T, Yorke JA, Casdagli M (1991) Embedology. J. Statistical Physics 65: 579-616
41. Sharifi MB, Georgakakos KP, Rodriguez-Iturbe I (1990) Evidence of deterministic chaos in the pulse of storm rainfall. J. Atmospheric Sci. 47(7): 888-893
42. Smith LA (1988) Intrinsic limits on dimension calculations. Physics Letters A 133: 283-288
43. Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds.) Dynamical Systems and Turbulence. Warwick, Lecture Notes Math. vol. 898, pp. 336-381
44. Tong H (1990) Non-linear Time Series: A Dynamical System Approach. Clarendon Press, Oxford
45. Tsonis AA (1992) Chaos: From Theory to Applications. Plenum Press, New York & London
46. Tsonis AA, Elsner JB (1988) The weather attractor over very short time scales. Nature 333: 545-547
47. Wilcox BP, Seyfried MS, Matison TH (1991) Searching for chaotic dynamics in snowmelt runoff. Water Resources Res. 27(6): 1005-1010
48. Zeng X, Eykholt R, Pielke RA (1991) Estimating the Lyapunov exponent spectrum from short time series of low precision. Physical Review Letters 66(25): 3229-3232